• Steel and Composite Structures
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• 제25권1호
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• pp.67-78
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• 2017

In this study, free vibration of functionally graded (FG) micro/nanobeams based on nonlocal third-order shear deformation theory and under different boundary conditions is investigated by applying the differential quadrature method. Third-order shear deformation theory can consider the both small-scale effects and quadratic variation of shear strain and hence shear stress along the FG nanobeam thickness. The governing equations are obtained by using the Hamilton's principle, based on third-order shear deformation beam theory. The differential quadrature (DQ) method is used to discretize the model and attain the natural frequencies and mode shapes. The properties of FG micro/nanobeam are assumed to be chanfged along the thickness direction based on the simple power law distribution. The effects of various parameters such as the nonlocal parameter, gradient index, boundary conditions and mode number on the vibration characteristics of FG micro/nanobeams are discussed in detail.

• Structural Engineering and Mechanics
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• 제6권7호
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• pp.727-746
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• 1998

Finite element models and numerical results are presented for bending and natural vibration using the unified third-order plate theory developed in Part 1 of this paper. The unified third-order theory contains the classical, first-order, and other third-order plate theories as special cases. Analytical solutions are developed using the Navier and L$\acute{e}$vy solution procedures (see Part 1 of the paper). Displacement finite element models of the unified third-order theory are developed herein. The finite element models are based on $C^0$ interpolation of the inplane displacements and rotation functions and $C^1$ interpolation of the transverse deflection. Numerical results of bending and natural vibration are presented to evaluate the accuracy of various plate theories.

• Structural Engineering and Mechanics
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• 제8권6호
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• pp.623-637
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• 1999

This paper presents a four-noded quadrilateral $C^0$ strain plate element for the analysis of thick laminated composite plates. The element formulation is based on: 1) the third-order shear deformation theory; 2) assumed strain element formulation; and 3) interrelated edge displacements and rotations along element boundaries. Unlike the existing displacement-type composite plate elements based on the third-order theory, which rely on the $C^1$-continuity formulation, the present plate element is of $C^0$-continuity, and its element stiffness matrix is evaluated explicitly. Because of the third-order expansion of the in-plane displacements through the thickness, the resulting theory and hence elements do not need shear correction factors. The explicit element stiffness matrix makes the present element more computationally efficient than the composite plate elements using numerical integration for the analysis of thick layered composite plates.

• Smart Structures and Systems
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• 제5권5호
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• pp.531-546
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• 2009

In the present analysis, the spline finite strip with higher-order shear deformation is formulated for the static analysis of piezoelectric composite plates. The proposed method incorporates Reddy's third-order shear deformation theory, Touratier's "Sine" model, Afaq's exponential model, Cho's higher-order zigzag laminate theory, as well as the classic plate theory and the first-order plate theory. Thus, the analysis can be conducted based on any of the above-mentioned theories. The selection of a specific method is done by simply changing a few terms in a 2 by 2 square matrix and the results, obtained according to different plate theories, can be compared to each other. Numerical examples are presented for piezoelectric composite plates subjected to mechanical loading. The results based on different shear deformation theories are compared with the three-dimensional solutions. The behaviours of piezoelectric composite plates with different length-to-thickness ratios, fibre orientations, and boundary conditions are also investigated in these examples.

• Structural Engineering and Mechanics
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• 제61권6호
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• pp.721-736
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• 2017

In this paper, free vibration characteristics of functionally graded (FG) nanobeams embedded on elastic medium are investigated based on third order shear deformation (Reddy) beam theory by presenting a Navier type solution for the first time. The material properties of FG nanobeam are assumed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on third order shear deformation beam theory through Hamilton's principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to some cases in the literature. The obtained results are presented for the vibration analysis of the FG nanobeams such as the influences of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

• Structural Engineering and Mechanics
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• 제27권1호
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• pp.1-16
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• 2007

In the present study, a spline finite strip with higher-order shear deformation is formulated for the stability and free vibration analysis of composite plates. The analysis is conducted based on Reddy's third-order shear deformation theory, Touratier's "Sine" model, Afaq's exponential model and Cho's higher-order zigzag laminate theory. Consequently, the shear correction coefficients are not required in the analysis, and an improved accuracy for thick laminates is achieved. The numerical results, based on different shear deformation theories, are presented in comparison with the three-dimensional elasticity solutions. The effects of length-to-thickness ratio, fibre orientation, and boundary conditions on the critical buckling loads and natural frequencies are investigated through numerical examples.

• Advances in nano research
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• 제6권2호
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• pp.93-112
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• 2018

An analytical solution of the buckling governing equations of functionally graded piezoelectric (FGP) nanobeams obtained by using a developed third-order shear deformation theory is presented. Electro-mechanical properties of FGP nanobeam are supposed to change continuously in the thickness direction based on power-law model. To capture the small size effects, Eringen's nonlocal elasticity theory is adopted. Employing Hamilton's principle, the nonlocal governing equations of a FG nanobeams made of piezoelectric materials are obtained and they are solved using Navier-type analytical solution. Results are provided to show the effect of different external electric voltage, power-law index, nonlocal parameter and slenderness ratio on the buckling loads of the size-dependent FGP nanobeams. The accuracy of the present model is verified by comparing it with nonlocal Timoshenko FG beams. So, this study makes the first attempt for analyzing buckling behavior of higher order shear deformable FGP nanobeams.

• Advances in nano research
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• 제5권4호
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• pp.313-336
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• 2017

This article investigates vibration behavior of magneto-electro-elastic functionally graded (MEE-FG) nanobeams embedded in two-parameter elastic foundation using a third-order parabolic shear deformation beam theory. Material properties of MEE-FG nanobeam are supposed to be variable throughout the thickness based on power-law model. Based on Eringen's nonlocal elasticity theory which captures the small size effects and using the Hamilton's principle, the nonlocal governing equations of motions are derived and then solved analytically. Then the influences of elastic foundation, magnetic potential, external electric voltage, nonlocal parameter, power-law index and slenderness ratio on the frequencies of the embedded MEE-FG nanobeams are studied.

• Earthquakes and Structures
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• 제26권4호
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• pp.299-309
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• 2024

The present paper develops application of third-order shear deformation theory (TSDT) and modified couple stress theory (MCST) to size-dependent bending analysis of a functionally graded cylindrical micro-shell. The radial and axial displacement components are described based on TSDT for more accurate analysis. The effect of small scales is accounted based on MCST. The principle of virtual work is used for derivation of bending governing equations. The solution is presented for a simply-supported boundary condition to account the influence of various important parameters such as micro length scale parameter, in-homogeneous index and some dimensionless geometric parameters such as length to radius and length to thickness ratios on the bending results. A comparative analysis is presented to examine the effect of order of employed shear deformation theory on the axial and radial displacements.

• 한국산학기술학회논문지
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• 제13권7호
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• pp.3207-3215
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• 2012

본 논문에서는 3차 전단변형이론이 고려된 비국소 탄성 이론을 이용한 나노 판의 휨 및 진동에 대하여 연구하였다. 비국소 탄성 이론은 미소 규모 효과를 고려할 수 있고 3차원 전단변형이론은 나노 판의 두께방향으로의 전단 변형률과 전단응력의 곡선변화 효과를 고려할 수 있다. 이러한 두 가지 이론을 이용하여 나노 판의 처짐과 고유진동수에 미치는 비국소 이론의 효과를 제시하였다. 국소 탄성이론과의 관계를 수치해석 결과를 통하여 고찰하였다. 또한 (i) 비국소 계수, (ii) 나노 판의 적층형태, (iii) 나노 판의 보강 방향 그리고 (iv) 나노 판의 적층 수 등이 나노 판의 무차원 처짐에 미치는 효과에 대하여 관찰하였다. 본 연구의 결과를 검증하기 위해 참고문헌의 결과들과 비교 분석하였으며 해석결과는 참고문헌의 결과들과 잘 일치함을 알 수 있었다. 비국소 이론에 의한 나노 판의 처짐에 관한 연구는 향후 관련연구에 비교자료로 활용될 수 있을 것이다.